D EPARTMENT OF P HYSICS , U MARU M USA YAR ’ ADUA U NIVERSITY, K ATSINA
PHY 101 General Physics I (Mechanics) Lecture Notes
n
io
ss
Se
4
Lecturers:
02
Shamsuddeen Sani Alhasan
/2
Hussaini Abubakar
Usman Sani
23
Jamaluddeen Kabir
Shamsu Muhammed Aliyu
Aliyu Lawal Albaba
20
Dr. Suleiman Bello
Bello Sa’adu
Saratu Abdulfatah
1
Dr. Nuraddeen Usman
0
Dr. Mahmud Abdulsalam
Y1
Prof. Bashir Gide Muhammad
Dr. Abdullahi Tanimu
PH
Dr. Yahaya Ibrahim (Course Co-ordinator)
2023/2024 Session
1
,Contents
C ONTENTS
1 Units and dimensions 3
1.1 The concept of a unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Fundamental Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Derived Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Unit Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 The concept of dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Vectors and scalar quantities 7
2.1 Vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Subtraction of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Components of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Unit vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
n
2.5 Product of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
io
3 Force, Momentum and the Newton Laws 11
3.1 Newton’s laws of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
ss
3.2 Momentum, Impulse, and Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.1 Conservation of linear momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Se
4 Work, Energy and power 15
4.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Total Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4
4.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . .
02 . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.1 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.2 Elastic Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3.3 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
/2
4.4 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.5 Conservative forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.5.1 Non-conservative forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
23
5 Rotational motion 20
5.0.1 Moment of inertia, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
20
5.0.2 Radius of gyration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.0.3 Angular momentum and its conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.0.4 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1
5.0.5 Kinetic energy of rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
0
6 Gravitation 22
Y1
6.1 Newton’s law of gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 Mass and weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.3 Gravitational potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
PH
6.4 Potential energy of a system of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.5 Satellites and escape velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.6 Kepler’s laws of planetary motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7 Projectiles 29
8 Circular motion 31
9 Elastic properties of solids 36
Recommended Textbooks
• Advanced Level Physics, Nelkon and Parker, CBS Publishers, India .
• University Physics, Sears and Zemansky, Pearson, (11th Edition).
2
,1 Units and dimensions
1 U NITS AND DIMENSIONS
1.1 T HE CONCEPT OF A UNIT
Assuming, a teacher measured the length of a wire and found it to be 20cm. Instead of writing 20cm, he
mistakenly wrote 20 only on the board. He then asked three of his students to tell him what was the length of
the ruler measured. As such, based on what was written on the board, student A said it was 20mm, student B
said it was 20m, student C said it was 20 inches. What brought about the different answers was the failure of
the teacher to attach a unit to the quantity measured. This has really illustrates an important point. Therefore,
quoting the result of a calculation or measurement without attaching a UNIT to it is useless. Frankly, this is a
common reason for many students losing marks in examinations. So, the concept of UNIT is of paramount
importance not only in physics. To make accurate and reliable measurements, we need units of measurement
that do not change and that can be duplicated by observers in various locations around the globe. The system
of units used by scientists and engineers around the world is commonly called "The metric system", but since
n
1960 it has been known officially as the International system of units, or SI (the abbreviation for its French
io
name is Systeme International). The main advantage of using a set of agreed units is that scientists from all
over the world can exchange ideas and designs for experiments without having to translate specifications into
different units. It is rather like having a common language.
ss
There are two types of units namely:
Se
• Base or Fundamental Units.
• Derived Units.
4
1.2
02
F UNDAMENTAL U NITS
Scientists have worked out that they can measure any quantity in nature in terms of a small number of base
/2
units. The challenge is to reduce the number of base/fundamental units to a minimum and still be able to
measure anything that can come up. Thus, it has been possible to keep to combinations of the seven base units
as listed below:
23
Quantity Name of Unit (SI base unit) Symbol
20
length meter m
mass kilogram kg
time second s
1
electric current ampere A
0
thermodynamic temperature kelvin K
amount of substance mole mol
Y1
luminous intensity candela cd
PH
NOTE: Any number that is used to describe a physical phenomenon quantitatively is called a physical quantity.
1.3 D ERIVED U NITS
From the word "derive", the derived units are obtained from combination of two or more fundamental units.
They enable us to measure more than the basic quantities of length, time, mass, etc. For instance, there is no
unit for speed among the base units. However, a suitable unit can be derived from the equation for speed as:
Distance (m)
Average speed = (1.1)
Time (s)
This suggests that the unit of speed is meters divide by seconds. Unfortunately, such a division is impossible.
Division can only happen when the quantities are of the same type. We now say instead that the unit of speed
3
, 1 Units and dimensions
is meters per second and denoted as ms −1 . Acceleration is another important quantity. It is the rate at which
speed is changing:
Final speed (ms −1 ) − Initial speed (ms −1 )
Average acceleration = (1.2)
Time (s)
The top line is the difference between two quantities in ms −1 . The bottom line is in seconds. Thus, the units of
acceleration are ’metre per second per second’, or ms −2 . The following table shows some other examples of the
derived units:
Quantity Derived unit Name
kg m s−2
n
force Newton (N)
pressure kg m−1 s−2 Pascal (Pa)
io
energy kg m2 s−2 Joule (J)
charge As Coulomb (C)
ss
volume m3 -
Se
1.4 U NIT P REFIXES
4
Once we have defined the fundamental units, it is easy to introduce larger and smaller units for the same
physical quantities. In the metric system, these other units are related to the fundamental units by multiples of
02
10 or 1/10. Thus, one kilometer (1 km) is 1000 meters, and one centimetre (1 cm) is 1/100 meter. We usually
express multiples of 10 or 1/10 in exponential notation; 1000 = 103 , 1/1000 = 10−3 and so on. With this notation,
1km = 103 m and 1cm = 10−2 m. The names of additional units are derived by adding a prefix to the name of
/2
the fundamental unit. For example, the prefix "kilo-," abbreviated k, always means a unit larger by a factor of
1000; thus
23
• 1 kilometer = 1km = 103 meters = 103 m.
• 1 kilogram = 1kg = 103 gram = 103 g.
20
• 1 kilowatt = 1 kW = 103 watts = 103 W.
The table below shows the standard prefixes (??):
0 1
1.5 T HE CONCEPT OF DIMENSION
Y1
Let consider a simple arithmetic equation such as:
PH
3+5 = 8
The equation balances, because the value of the numbers on the left-hand side is equal to the value of numbers
on the right-hand side. In physics, equations usually equate quantities that have magnitude (values), dimen-
sions and units. All three must balance for the equation to be meaningful.
However, the word dimension has a special meaning in physics. It denotes the physical nature of a quantity.
Whether a distance is measured in units of feet or metres or inches, it is still a distance. We say its dimension is
length.
In other words, dimension is the type of quantity we are dealing with independent of its units or value. For
instance, 100cm, 1m, 2 miles and 3 light-years all have the dimension of length, but are expressed in different
units. Simple numbers, as in the above arithmetic equation, are dimensionless, whereas dimensions are
valueless. The examples below show situations in which values balance but dimensions and/or units do not.
4
PHY 101 General Physics I (Mechanics) Lecture Notes
n
io
ss
Se
4
Lecturers:
02
Shamsuddeen Sani Alhasan
/2
Hussaini Abubakar
Usman Sani
23
Jamaluddeen Kabir
Shamsu Muhammed Aliyu
Aliyu Lawal Albaba
20
Dr. Suleiman Bello
Bello Sa’adu
Saratu Abdulfatah
1
Dr. Nuraddeen Usman
0
Dr. Mahmud Abdulsalam
Y1
Prof. Bashir Gide Muhammad
Dr. Abdullahi Tanimu
PH
Dr. Yahaya Ibrahim (Course Co-ordinator)
2023/2024 Session
1
,Contents
C ONTENTS
1 Units and dimensions 3
1.1 The concept of a unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Fundamental Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Derived Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Unit Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 The concept of dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Vectors and scalar quantities 7
2.1 Vector addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Subtraction of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Components of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Unit vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
n
2.5 Product of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
io
3 Force, Momentum and the Newton Laws 11
3.1 Newton’s laws of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
ss
3.2 Momentum, Impulse, and Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.1 Conservation of linear momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Se
4 Work, Energy and power 15
4.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Total Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4
4.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . .
02 . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.1 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.2 Elastic Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3.3 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
/2
4.4 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.5 Conservative forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.5.1 Non-conservative forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
23
5 Rotational motion 20
5.0.1 Moment of inertia, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
20
5.0.2 Radius of gyration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.0.3 Angular momentum and its conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.0.4 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1
5.0.5 Kinetic energy of rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
0
6 Gravitation 22
Y1
6.1 Newton’s law of gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 Mass and weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.3 Gravitational potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
PH
6.4 Potential energy of a system of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.5 Satellites and escape velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.6 Kepler’s laws of planetary motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7 Projectiles 29
8 Circular motion 31
9 Elastic properties of solids 36
Recommended Textbooks
• Advanced Level Physics, Nelkon and Parker, CBS Publishers, India .
• University Physics, Sears and Zemansky, Pearson, (11th Edition).
2
,1 Units and dimensions
1 U NITS AND DIMENSIONS
1.1 T HE CONCEPT OF A UNIT
Assuming, a teacher measured the length of a wire and found it to be 20cm. Instead of writing 20cm, he
mistakenly wrote 20 only on the board. He then asked three of his students to tell him what was the length of
the ruler measured. As such, based on what was written on the board, student A said it was 20mm, student B
said it was 20m, student C said it was 20 inches. What brought about the different answers was the failure of
the teacher to attach a unit to the quantity measured. This has really illustrates an important point. Therefore,
quoting the result of a calculation or measurement without attaching a UNIT to it is useless. Frankly, this is a
common reason for many students losing marks in examinations. So, the concept of UNIT is of paramount
importance not only in physics. To make accurate and reliable measurements, we need units of measurement
that do not change and that can be duplicated by observers in various locations around the globe. The system
of units used by scientists and engineers around the world is commonly called "The metric system", but since
n
1960 it has been known officially as the International system of units, or SI (the abbreviation for its French
io
name is Systeme International). The main advantage of using a set of agreed units is that scientists from all
over the world can exchange ideas and designs for experiments without having to translate specifications into
different units. It is rather like having a common language.
ss
There are two types of units namely:
Se
• Base or Fundamental Units.
• Derived Units.
4
1.2
02
F UNDAMENTAL U NITS
Scientists have worked out that they can measure any quantity in nature in terms of a small number of base
/2
units. The challenge is to reduce the number of base/fundamental units to a minimum and still be able to
measure anything that can come up. Thus, it has been possible to keep to combinations of the seven base units
as listed below:
23
Quantity Name of Unit (SI base unit) Symbol
20
length meter m
mass kilogram kg
time second s
1
electric current ampere A
0
thermodynamic temperature kelvin K
amount of substance mole mol
Y1
luminous intensity candela cd
PH
NOTE: Any number that is used to describe a physical phenomenon quantitatively is called a physical quantity.
1.3 D ERIVED U NITS
From the word "derive", the derived units are obtained from combination of two or more fundamental units.
They enable us to measure more than the basic quantities of length, time, mass, etc. For instance, there is no
unit for speed among the base units. However, a suitable unit can be derived from the equation for speed as:
Distance (m)
Average speed = (1.1)
Time (s)
This suggests that the unit of speed is meters divide by seconds. Unfortunately, such a division is impossible.
Division can only happen when the quantities are of the same type. We now say instead that the unit of speed
3
, 1 Units and dimensions
is meters per second and denoted as ms −1 . Acceleration is another important quantity. It is the rate at which
speed is changing:
Final speed (ms −1 ) − Initial speed (ms −1 )
Average acceleration = (1.2)
Time (s)
The top line is the difference between two quantities in ms −1 . The bottom line is in seconds. Thus, the units of
acceleration are ’metre per second per second’, or ms −2 . The following table shows some other examples of the
derived units:
Quantity Derived unit Name
kg m s−2
n
force Newton (N)
pressure kg m−1 s−2 Pascal (Pa)
io
energy kg m2 s−2 Joule (J)
charge As Coulomb (C)
ss
volume m3 -
Se
1.4 U NIT P REFIXES
4
Once we have defined the fundamental units, it is easy to introduce larger and smaller units for the same
physical quantities. In the metric system, these other units are related to the fundamental units by multiples of
02
10 or 1/10. Thus, one kilometer (1 km) is 1000 meters, and one centimetre (1 cm) is 1/100 meter. We usually
express multiples of 10 or 1/10 in exponential notation; 1000 = 103 , 1/1000 = 10−3 and so on. With this notation,
1km = 103 m and 1cm = 10−2 m. The names of additional units are derived by adding a prefix to the name of
/2
the fundamental unit. For example, the prefix "kilo-," abbreviated k, always means a unit larger by a factor of
1000; thus
23
• 1 kilometer = 1km = 103 meters = 103 m.
• 1 kilogram = 1kg = 103 gram = 103 g.
20
• 1 kilowatt = 1 kW = 103 watts = 103 W.
The table below shows the standard prefixes (??):
0 1
1.5 T HE CONCEPT OF DIMENSION
Y1
Let consider a simple arithmetic equation such as:
PH
3+5 = 8
The equation balances, because the value of the numbers on the left-hand side is equal to the value of numbers
on the right-hand side. In physics, equations usually equate quantities that have magnitude (values), dimen-
sions and units. All three must balance for the equation to be meaningful.
However, the word dimension has a special meaning in physics. It denotes the physical nature of a quantity.
Whether a distance is measured in units of feet or metres or inches, it is still a distance. We say its dimension is
length.
In other words, dimension is the type of quantity we are dealing with independent of its units or value. For
instance, 100cm, 1m, 2 miles and 3 light-years all have the dimension of length, but are expressed in different
units. Simple numbers, as in the above arithmetic equation, are dimensionless, whereas dimensions are
valueless. The examples below show situations in which values balance but dimensions and/or units do not.
4
